Problem: All of the 3rd grade teachers and students from Almond went on a field trip to an archaeology museum. Tickets were $$5.50$ each for teachers and $$2.50$ each for students, and the group paid $$33.50$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$16.50$ each for teachers and $$12.00$ each for students, and the group paid $$141.00$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5.5x+2.5y = 33.5}$ ${16.5x+12y = 141}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-16.5x-7.5y = -100.5}$ ${16.5x+12y = 141}$ Add the top and bottom equations together. $ 4.5y = 40.5 $ $ y = \dfrac{40.5}{4.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {5.5x+2.5y = 33.5}$ to find $x$ ${5.5x + 2.5}{(9)}{= 33.5}$ $5.5x+22.5 = 33.5$ $5.5x = 11$ $x = \dfrac{11}{5.5}$ ${x = 2}$ You can also plug ${y = 9}$ into $ {16.5x+12y = 141}$ and get the same answer for $x$ ${16.5x + 12}{(9)}{= 141}$ ${x = 2}$ There were $2$ teachers and $9$ students on the field trips.